FOM: F.O.M./pure math; general intellectual interest

[Harvey Friedman]

Lou writes:

>Concerning Harvey's "needle in the haystack comment". It was not
>my intent to argue that position, and I am sorry my comment about
>"who, 200 years ago, could have predicted ..." was a bit elliptic.
>
>What i meant to say is that central ideas often emerge only with
>great difficulty and passing of time, even while not at all difficult
>by themselves. They just do not lie on the surface. Elliptic
>curves y^2 = x^3 + ax + b are not wildly complicated things compared
>to the objects considered typically in fom research, nor are
>zeta functions. Zeta functions (of arithmetic objects) seem to
>encode in an uncanny way an enormous amount of information about
>the object, and often satisfy remarkable symmetries, functional
>equations, etc.

I agree completely. But by saying this, you give the unfair impression that
I would disagree with this.

>You can denounce it as "inbred" and "snooty",
>it doesn't take away the facts.

No. I denounced as "inbred" and "snooty" many aspects of the current pure
mathematical culture

>While I am far from being an expert,
>I think it is very possible that the uses of such objects in
>some complexity questions is not at all temporary or an accident.

The number theory comes in mainly as a way of providing constructions that
is easy to make, yet seemingly difficult to reverse. E.g., even if you know
that n is the product of two primes, it is somewhat believed, but by no
means established, that it is difficult to find the factors given the
product. Trouble is, there is perhaps no good reason to believe that it is
difficult to find these factors, particularly since elementary number
theory is so hard and striking advances are made unexpectedly from time to
time. Therefore the search is on by many important reserachers to replace
it by more basic, more generic problems that look difficult, and to prove
"equi-hard" theorems of the appropriate kind. E.g.,
1. Generating Hard Instances of Lattice Problems, extended abstract, M. Ajtai.
2. A public-key crptosystem with worst-case/average-case equivalence,
extended abstract, Miklos Ajtai and Cynthia Dwork.

Thus the reliance on the number theory grows out of the fact that number
theory is perceived to be difficult, more than any intrinsic property of
number theory - or so it seems.

The point is, if you want to claim competitively high general intellectual
interest for contemporary number theory by citing its applications to
cryptography, well: G-- help you. You'll need Him.

>In any case, I think the burden to show otherwise is on your side.

No, given the way that it is used as indicated above.

> Also note that I am not a number theorist or complexity theorist
>(while having some interest in those areas), and have no plans to
>work on P=NP? (having other fish to fry) using elliptic curves
>or whatever.

Well, I asked you to consider a comparison between P=NP and what you
routinely get so excited about in pure mathematics. How big are your fish?

>Still, I don't think it's a treasonable offense to
>know a little bit on elliptic curves or zeta functions. I am
>amazed how these poor things seem to elicit so much abuse on
>mathematicians.

I agree completely. But by saying this, you give the unfair impression that
I would disagree with this.

Lou - I picked a fight with you in 5:21PM 12/15/97. Looks like you want to
walk off the playground, cuddling your zeta functions and eliptic curves.
Anybody else care to fight?